Preliminary study of the Gravimetric Local Geoid Model in Jordan

ANNALS OF GEOPHYSICS, VOL. 50, N. 3, June 2007
Preliminary study of the Gravimetric
Local Geoid Model in Jordan:
case study (GeoJordan Model)
Omar Al-Bayari and Abdallah Al-Zoubi
Department of Surveying and Geomatics Engineering, Faculty of Engineering,
Al-Balqa’ Applied University, Al-Salt, Jordan
Abstract
Recently, there has been an increased interest in studying and defining the Local and Regional Geoid Model worldwide, due to its importance in geodetic and geophysics applications.The use of the Global Positioning System (GPS)
is internationally growing, yet the lack of a Geoid Model for Jordan has limited the use of GPS for the geodetic applications in the country. This work aims to present the preliminary results that we propose for «The Gravimetric Jordanian Geoid Model (GeoJordan)». The model is created using gravimetric data and the GRAVSOFT program. The
model is validated using GPS and precise level measurements in the Amman area. Moreover, we present a comparison using the Global Geopotential Model OSU91A and the EGM96 Model and the results showed great discrepancies. We also present the approach used to obtain the orthometric height from GPS ellipsoidal height measurements.
We found that the error margin obtained in this work of the GeoJordan after fitting the data with GPS/leveling measurements is about (10 cm) in the tested area whereas the standard error of the created model is about (40 cm).
2002; Fotopoulos, 2003). Therefore, it became
indispensable to define the geoid undulation in
Jordan to allow more usage and benefit of GPS in
the area. Furthermore, the determination of the
Geoid Model for Jordan never was easy due to:
the high costs of the process, difficulty accessing
the gravimetric data that covers the country, difficulty to have GPS and precise leveling for the
country to end with, the lack of access to the
gravimetric data in the neighboring countries.
On the other hand, we used available gravimetric data collected by the Natural Resources
Authority of Jordan (NRAJ) for geophysical purposes (Al-Zoubi, 2002). The number of gravimetric points measured offered a possibility to start
creating a Geoid Model for Jordan. To validate
the Gravimetric Geoid Model, GPS and precise
leveling measurements were performed by the
authors (at the Surveying and Geomatics Engineering department/Al-Balqa’ Applied University) over a small area. The SRTM Digital Elevation Model (DEM) was used for the reduction of
Key words Gravimetric Jordanian Geoid Model
(GeoJordan) – Least Square Collocation (LSC) –
Global Geopotential Model (GGM) – Global Positioning System (GPS) – undulation (N)
1. Introduction
The height obtained using GPS satellitebased-system is referred to the ellipsoid. Many
engineering and geodetic applications are interested in the orthometric height, which is the
height above the geoid, not above the ellipsoid
(Featherstone et al., 1998). Thus the transformation between the two heights could be obtained
directly knowing the geoid undulations (Marti,
Mailing address: Dr. Omar Al-Bayari, Department of
Surveying and Geomatics Engineering., Faculty of Engineering, Al-Balqa’ Applied University, Al-Salt 19117, Jordan; e-mail: [email protected]
387
Omar Al-Bayari and Abdallah Al-Zoubi
Fig. 1. The relationship between orthometric, geoid and ellipsoidal heights.
the gravimetric data (Kiamehr and Sjoberg,
2005). In some areas more accurate DEM (produced by the authors from stereo SPOT images
via digital photogremmetry techniques) was used.
The Geoid Model could be obtained by GPS/
leveling measurements (geometric method)
(Duquesne et al., 1995; Fotopoulos, 2003), or the
gravimetric method (Rapp, 1997; Featherstone
et al., 2001). While the geometric method is not
easy to implement due to the poor spatial coverage of geometric leveling lines (Lee and Mazera,
2000), the gravimetric method utilizes a better
distribution of terrestrial gravity observations and
a global geopotential model (Bottoni and Barzaghi, 1993; Amos and Featherstone, 2003). Moreover, the geoid is considered to be a reference for
the Earth gravity field and/or represents the vertical datum that permits the study of the sea-level.
The geometric relation between the geoid, ellipsoid and Earth surface is shown in fig. 1, where
the separation between orthometric height (H)
and ellipsoidal height (h) is known as the geoid
undulation (N) (Heiskanen and Moritz, 1967)
N = h−H.
tric heights (H), (Al-Bayari et al., 1996) providing a benefit of utilizing GPS measurements as
an alternative of the precise leveling for many
applications.
2. Methodology used and Global
Geopotential Model
There are many procedures that may be used
for geoid determination (Sideris and She, 1995;
Barzaghi et al., 1996; Arabelos and Tscherning,
2002). However, in this work the GeoJordan
model is determined by the remove-restore and
Least-Squares Collocation (LSC) procedure,
implemented in GRAVSOFT package (Tscherning, 1994). The remove-restore approach is utilized whenever the long and short wavelength
components of the geoid are computed. The
main steps are:
– Spherical harmonic expansion impact; the
remove procedure is carried out by computing the
long wavelength component as gravity anomalies
from the Global Geopotential Model (∆gM), then
subtracted from the raw gravity (∆g). This step is
done for the two GGM’s OSUA91A and EGM96
for analysis purposes (GEOCOL program).
– Residual Terrain Model (RTM) impacts;
the short wavelength geoid height is then comput-
(1.1)
The initial practical application of the geoid undulation (N) in land surveying is to transform
GPS- derived ellipsoidal heights (h) to orthome388
Preliminary study of the Gravimetric Local Geoid Model in Jordan: case study (GeoJordan Model)
GM
a n
NM = rγ / b r l / (δCr nm cos mλ + Srnm sin mλ) $
n=2
m=0
$ Prnm (cos θ)
M
ed from the resulting residual anomalies (∆gRTM).
The terrain correction program in GRAVSOFT
package (TC program) was used to compute the
effect of topography from the Digital Terrain
Model (DTM). The DTM is handled by analytical prism integration assuming a constant density
of all masses above sea-level. After that the long
wavelength component is restored as a geoid
height computed from the GGM.
– The final step is to add the two components,
short and long wavelength, producing a geoid
height model. LSC is used to estimate the geoid
heights and their errors (GEOCOL program).
In the remove-restore procedure the undulation (N) is split into three components
N = NM + NRTM + Nr .
whereas GM is the product of the Newtonian
gravitational constant and mass of the Earth, γ
is normal gravity on the surface of the reference ellipsoid, (r, θ, λ) are the geocentric spherical polar coordinates of the point at which ∆g
is to be determined; a is the semi-major axis of
the geocentric reference ellipsoid; δC⎯ nm and
S⎯ nm are the fully normalized spherical geopotential coefficients of the GGM, reduced; P
⎯ nm
are the fully normalized associated Legendre
functions for degree n and order m; and M is
the maximum degree of spherical harmonic expansion.
OSU91A GGM model does not have enough
data from the Middle East, but the EGM96 model contains some data from the Middle East.
Therefore we took the EGM96 as the base model to represent the final results in this study. Also
we have to consider that the data used to realize
the EGM96 is highly compatible with new reference systems, such as ITRF.
(2.1)
Where NM represents the contribution from the
GGM, NRTM the residual terrain effect contribution and Nr the residual anomaly filed (∆gr), after
removal of the contribution of GGM and terrain
effect or the contribution of residual gravity
anomalies
∆gr = ∆g − ∆gM − ∆gRTM .
(2.2)
3. Data used for Gravimetric Jordanian
Geoid Model
2.1. Global Geopotential Models
Usually, the Global Geopotential Model
(GGM) is computed as a series of spherical harmonic expansions to a maximum degree and sort
comprises that describe the long-wavelength
characteristics of the Earth’s gravity field.
OSU91A (Rapp et al., 1991) and EGM96 (Lemoine et al., 1998) are the most common global
geopotential models applied for the Geoid Model computations. However, these models are
completed up to degree and order 360 gravity
anomalies and can be computed in spherical approximation from the geopotential coefficients
(Rapp, 1997)
Jordan extends between latitudes 29° to 33°
and longitudes 35° to 39° and the area of Jordan
is 87000 km2. The Gravimetric Jordanian Geoid
Model uses a combination of three input data
sources:
– Gravity data are collected by the NRAJ
mainly for geophysical purposes; distributed
over the whole country (about 3000 free-air
gravity anomalies), they allow to determine the
effect of the intermediate geoid wave length,
around 5 to 10 km. These data are part of the
database covering the Jordan territory and are
referenced to IGSN71. The distribution of the
gravity data used is shown in fig. 2, the distance between points is approximately 5 km,
and the standard error declared by NRAJ is 2
mgal.
– Global Geopotential Model (GGM) to
determine the long wave length of geoid undulations more than 100 km.
GM b a ln
(δCr nm cos mλ +
r (n − 1) /
r2 /
n=2
m=0
+ Srnm sin mλ) Prnm (cos θ)
∆g M =
M
n
n
and the geoid height is computed from applying
the GGM
389
Omar Al-Bayari and Abdallah Al-Zoubi
– Digital Elevation Model (DEM) which
supplies most of the short wavelengths (∼100 m)
and is also required to satisfy theoretical demands
of geoid computation from the geodetic boundary-value problem. The resolution of SRTM used
is 3m×3m pixel size and extended between 29°≤
≤ ϕ ≤ 33° and 35° ≤ λ ≤ 39°. The SRTM DEM is
edited using Terrascan and Terramodel programs
(modules working under Microstation softwareBentley). Furthermore, the DEM shows good
agreement with the topographic maps (differences less than 20 m). The DEM used in the study
area was 20 m pixel size and is produced using
PCI-Geomatica software using SPOT-5 images;
the standard error obtained of this DEM is less
than 10 m.
Since the normal gravity was evaluated on the
surface of the GRS80 ellipsoid at the geocentric
latitude of the gravity observation using Somigliana’s closed formula (Moritz, 1980), therefore,
the effect of vertical datum should be considered
due to different reference surfaces used in rawdata; and to be aware of the transformation of
GPS height to the local vertical datum. It is important to point out that all coordinate transformation from the national coordinate system to the
geocentric datum was performed by applying a
locally developed program.
Thus, the raw-data has been checked in the
preprocessing step via various interpolation
routines to isolate and eliminate error points.
Error points elimination is very important because the errors will directly propagate into the
created Geoid Model (Tscherning, 1991). Then,
about 100 gravity data were considered to be
outliers during preprocessing.
Fig. 2. Gravity data distribution over Jordan territory (latitude and longitude in WGS84).
4. GPS/leveling measurements
The GPS/leveling measurements were carried
out to study the accuracy of the Gravimetric
Geoid Model (GeoJordan) (Kotsakis and Sideris,
1999). Two GPS observation techniques was
used: the static and the rapid static techniques.
The GPS static measurements were used to establish the reference network in the tested area using
Leica SR530 GPS dual frequency receivers. The
data were processed using Bernese 4.2 software
and the static points were used as reference for
Fig. 3. GPS Static reference points and rapid static
points (latitude and longitude in WGS84).
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Preliminary study of the Gravimetric Local Geoid Model in Jordan: case study (GeoJordan Model)
rapid static survey. The rapid static and leveling
measurements were used for the determination of
geoidal undulation in the vicinity of Spirit leveling network in the study area (fig. 3). SKI-Pro
software was used for GPS data processing and
the accuracy obtained is within 3 cm, considering
that the baselines lengths are restricted to 5 km.
Spirit leveling network in Jordan is determined
with respect to the mean sea-level defined at Aqaba Gulf.
5. Results
Actual computation of GeoJordan was done
in accordance with the GRAVSOFT program using GGM models OSU91A and EGM96; accordingly the results are shown in fig. 4, and any error
associated with this model is presented in fig. 5
where the standard error in flat areas is 0.2 m and
reaches to 1 m in mountainous areas. However,
this error resulted from the lack of gravity data
particularly at the mountainous areas.
Fig. 5. The standard error associated with the
«GeoJordan» (latitude and longitude in WGS84 and
errors are expressed in m).
Table I. Statistical parameters for the gravity residual
(∆gr) computed using EGM96 and using OSU91A.
Statistical parameters
OSU91A
(mGal)
EGM96
(mGal)
No. of points
Average
Min
Max
Standard deviation
2994
−5.91
−211.33
102.60
31.82
2994
−2.52
−162.82
86.39
28.44
Statistical analysis of the gravity residual
(∆gr) computed using EGM96 (table I) shows
better behavior than those computed using
OSU91A, and the differences range from 1 to 2
m (fig. 6). Then, the EGM96 is more accurate
than the OSU91A due to the lack of gravity data of the Middle East area within the OSU91A
model and to the long wavelength error propagation in the GGM.
Fig. 4. The Gravimetric Jordanian-Geoid-Model
«GeoJordan» (latitude and longitude in WGS84 and
geoidal undulation in m).
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Omar Al-Bayari and Abdallah Al-Zoubi
Fig. 6. The differences in meters, between the two models created in Jordan using OSU91A and EGM9 (latitude and longitude in WGS84).
urements, the nominal resulting accuracy was at
decimeter level, these values were deducted by
interpolating the model by means of weighted average technique (Yanalak and Baykal, 2001).
Thus the point differences between the GPS undulation and the GeoJordan model showed a systematic behavior in some zones (fig. 7). However,
the short wavelength differences may have resulted from the following errors: error in GPS/ leveling data, or due to localized errors in the gravity
or terrain data. The topographic effect within the
tested area could be the main factor for these differences (fig. 8). To model this systematic behavior and to decrease the impact of the vertical datum shift, a best-fit regression plane and multiquadric interpolation were used, where GPS data
are assumed as reference (x, y, and N). Therefore,
both regression and multiquadric are subtracted
from the GeoJordan undulation (fig. 9).
The mathematical models for regression
plane and multiquadric interpolation are reported in (Yanalak and Baykal, 2001)
6. Model validation and discussion
The experimental Geoidal undulation was
calculated via a high quality GPS baseline,
which is a promising approach in a local area
due to its reliability, accuracy and even cost/
benefit ratio, especially for engineering and hydraulic applications. Meanwhile the experimental Geoidal surface and contour maps were created by interpolating the undulation values in a
specific study area (in Amman city) to compare
the experimental model with our resulting GeoJordan model.
In a first attempt: the experimental undulation in the tested area was compared with GeoJordan created from gravity data and the GGM
OSU91A. However, as expected the agreement
between GPS derived data and this model were
not high-quality in the tested area.
In a second attempt: the GPS derived geoid
was compared to the GeoJordan model created
with GGM EGM96. At the points of GPS meas392
Preliminary study of the Gravimetric Local Geoid Model in Jordan: case study (GeoJordan Model)
Fig. 7. Differences in meters, between Experimental Undulation (GPS/Levelling data) and gravimetric Jordananian Geoid Model (GeoJordan) before fitting (latitude and longitude in WGS84).
Fig. 8. The DTM extracted from the topographic map in tested area (latitude and longitude in WGS84 and contours value in meters).
393
Omar Al-Bayari and Abdallah Al-Zoubi
Nexp − NJordanGeo = a00 + a10 .x + a01 .y
(6.1)
z (x0, y0) = a00 + a01 .y0 + a10 .x0 + a02 . (x0) 2 +
+ a11 .x0 .y0 + a02 . (y0) 2
(6.2)
Nexp − NJordanGeo = z (x0, y0) + / c j 6(x j − x0) 2 +
the remaining points. The method to fit the
geoid’s model with the experimental undulation
is implemented in the C++ program. The program is also used to transform the ellipsoidal
height into orthometric height with their errors as
reported in (Yanalak and Baykal, 2001).
The statistical results of the comparison are
presented in table II, and the differences between
the GeoJordan model and the experimental undulation after the application of the fitting procedure
are presented in fig. 9. Since little information on
the quality of leveling data and the vertical datum
used in Jordan are available, the fitting method
will minimize the effect of datum shift between
the gravimetric model and geometric model.
Table II shows that the multiquadric fitting has
better behavior than regression plane due to irregular topography in the study area (fig. 8). This
will be noticed when the tested area is extended
in the near future using GPS/leveling measurements. The discrepancies appearing in fig. 7
(mainly due to the topography at the study area
(fig. 8), force us to improve the model by includ-
m
+ (y j − y0) @
j=1
(6.3)
2 0.5
∆z j = (Nexp − NJordanGeo) − z (x0, y0) .
(6.4)
Where (a00 ... a02) are the coefficient of the polynomial function that articulate the surface; in
multiquadric interpolation cj is the unknown to be
determined using the residuals ∆zj at known reference points (xj, yj). The least square method is
used for the coefficient determination.
Naturally, the reference points should have a
homogeneous distribution on the study area and
should be available for interpolation of most of
Fig. 9. Differences in meters, between Experimental Undulation (GPS/Leveling data) and gravimetric Jordanian Geoid Model (GeoJordan) after fitting (latitude and longitude in WGS84).
394
Preliminary study of the Gravimetric Local Geoid Model in Jordan: case study (GeoJordan Model)
Table II. Statistical parameters for the differences between WGS84 and GeoJordan Model (corrected by a regression plane and multiquadric).
Amman area (tested area)
Statistical parameters
GPS/Leveling measurements
for the difference between
and GeoJordan
WGS84 and GeoJordan Model
Average
Min
Max
Standard deviation
−1.15
−1.85
0.25
0.41
The GeoJordan
(corr. by regression)
(m)
The GeoJordan
(corr. by multiquadric)
(m)
−0.11
−0.43
0.45
0.19
0.01
-0.37
0.24
0.16
Due to the lack of access to the gravimetric
data in the neighboring countries the border effect is clearly shown in the created model. To
overcome this problem in the near future a regional model could be used instead of the global model, or the existing gravity data be combined with long-wave spectral components of
the global model using a least squares spectral
combination. This could be done by collaboration with national and international centers,
such as International Geoid Service (IGeS).
This work also showed that the multiquadric
and regression plane interpolation methods
could be used for a local height transformation
problem of GPS at the level of decimeter accuracy for a region with an area of about 40-50 km2.
Finally, the multiquadric interpolation gives
a good level of accuracy in the extrapolation
process, particularly in the irregular topographic areas.
ing more gravity data and/or GPS/leveling data in
the LSC procedure to create a centimeter Geoid
Model. Consequently the fitting model shown in
fig. 9 will not satisfy all the GPS users to obtain
a reliable orthometric height at centimeter level
from GPS measurements.
The second attempt results (table II and fig. 9)
showed good correlation between GPS derived
geoids and GeoJordan. This confirms the reliability and the high potential of GPS measurements
combined with spirt leveling to adjust the «GeoJordan» particularly in irregular and mountainous
areas.
7. Conclusions
A preliminary result of the Jordanian Gravimetric Geoid Model the «GeoJordan» is presented in this paper.
The model was validated through a comparison with a measured GPS/leveling data at Amman area and showed fairly good results.
The comparison also showed a high frequency problem in the estimation of GeoJordan, consequently this model needs more effort
and time to be more accurate. The database
should also be improved and extended to improve accuracy. This improvement includes: increasing the number and density of points (new
GPS surveys are currently underway), combining the datasets with gravimetric data and definition of the vertical datum used in Jordan for
orthometric height. This will be a great advantage in the analysis of results.
REFERENCES
AL-BAYARI, O., G. BITELLI, A. CAPRA, D. DONIMICI, E. ERCOLANI, G. FOLLONI, S. GANDOLFI, A. PELLEGRINELLI,
M. UNGUENDOLI and L. VITTUARI (1996): A local geoid
in the South-East of the Po Valley, in Proceedings of
the XXI Assembly European Geophysical Society, 6-10
May 1996, The Hague, The Netherlands, Session G-7
‘Techniques for local geoid determination’, 157-163.
AL-ZOUBI, A.S. (2002): The Dead Sea Basin, its structural
setting and evaporite tectonics, plate boundary zones,
Geodyn. Ser. 30, 10.1029/030GD09, 145-172.
AMOS, M.J. and W.E. FEATHERSTONE (2003): Progress towards a gravimetric geoid for New Zealand and a single
national vertical datum, in Proceedings of the 3rd Meeting of the International Gravity and Geoid Commission,
395
Omar Al-Bayari and Abdallah Al-Zoubi
Gravity and Geoid 2002-GG2002, Thessaloniki, Greece,
395-400.
ARABELOS, D. and C.C. TSCHERNING (2003): Globally covering apriori regional gravity covariance models, Adv.
Geosci., 1, 143-147.
BARZAGHI, R., M.A. BROVELLI, A. MANZINO, D. SGUERSO and
G. SONA (1996): The new Italian quasi-geoid ITALGEO95, Boll. Geod. Sci. Affini, 15 (1), 57-72.
BOTTONI, G.P. and R. BARZAGHI (1993): Fast collocation,
Bull. Geod., 67, 119-126.
DUQUENNE, H., Z. JIANG and C. LEMARIE (1995): Geoid determination and levelling by GPS: some experiments
on a test network, in Proceedings of the IAG Symposium, 1994, Graz, Austria, 559-568.
FEATHERSTONE, W.E., M.S. DENITH and J.F. KIRBY (1998):
Strategies for the accurate determination of Orthometric Heights, Surv. Rev., 34 (267), 278-296.
FEATHERSTONE, W.E., J.F. KIRBY, A.H.W KEARSLEY, J.R
GILLILAND, G.M. JOHNSTON, J. STEED, R. FORSBERG and
M.G. SIDERIS (2001): The AUSGeoid98 Geoid Model of
Australia: data treatment, computations and comparisons
with GPS-levelling data, J. Geod., 75, 313-330.
FOTOPOULOS, G. (2003): An analysis on the optimal combination of Geoid, orthometric and ellipsoidal height data, Ph.D. Thesis (University of Calgary, Department of
Geomatics Engineering, Canada).
HEISKANEN, W.A. and H. MORITZ (1967): Physical Geodesy
(W.H. Freeman and Company San Francisco), pp. 370.
KIAMEHR, R. and L.E. SJOBERG (2005): Effect of the SRTM
global DEM in the determination of a high-resolution
Geoid Model of Iran, J. Geod., 79 (9), 540-551.
KOTSAKIS, C. and M.G. SIDERIS (1999): On the adjustment
of combined GPS/levelling/geoid networks, J. Geod.,
73 (8), 412-421.
LEE, J.T. and D.F. MAZERA (2000): Concerns related to GPSderived Geoid determination, Surv. Rev., 35 (276), 379397.
LEMOINE, F.G., S.C. KENYON, R.G. FACTOR, R.G. TRIMMER,
N.K. PAVLIS, D.S. CHINN, C.M. COX, S.M. KLOSKO,
S.B. LUTHCKE, M.H. TORRENCE, Y.M. WANG, R.G.
WILLIAMSON, E.C. PAVLIS, R.H. RAPP and T.R. OLSON
(1998): The development of the joint NASA GSFC and
the National Imagery and Mapping Agency (NIMA)
geopotential model EGM96, NASA/TP-1998-206861
(NASA, Washington), pp. 575.
MART, U. (2002): Modelling of different height systems in
Switzerland, presented at the IAG Third Meeting of International Gravity and Geoid Commission, August
26-30, 2002, Thessaloniki, Greece.
MORITZ, H. (1980): Geodetic reference system 1980, Bull.
Geod., 54, 395-405.
RAPP, R.H. (1997): Use of the potential coefficient models
for geoid undulation determinations using harmonic
representation of the height anomaly/geoid undulation
difference, J. Geod., 71 (5), 282-289.
RAPP, R.H., Y.M. WANG and N.K. PAVLIS (1991): The Ohio
State 1991 geopotential and sea surface topography
harmonic coefficient model, Report 410 (Department
of Geodetic Science and Surveying, Ohio State University, Columbus).
SIDERIS, M. and B. SHE (1995): A new, high-resolution
geoid for Canada and part of the US by the 1D-FFT
method, Bull. Geod., 69 (2), 92-108.
TSCHERNING, C.C. (1991): The use of optimal estimation for
gross-error detection in databases of spatially correlated data, BGI, Bull. Inf., 68, 79-89.
TSCHERNING, C.C. (1994): Geoid determination by leastsquares collocation using GRAVSOFT, Lecture Notes of
the International School for the Determination and Use
of the Geoid (DIIAR - Politecnico di Milano, Milano).
YANALAK, M. and O. BAYKAL (2001): Transformation of ellipsoid heights to local leveling heights, J. Surv. Eng.,
127 (3), 90-103.
(received January 19, 2007;
accepted May 3, 2007)
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